Monday, May 27, 2013

Bets do not (necessarily) reveal beliefs

Bryan Caplan is well-known for demanding that people bet on their macroeconomic beliefs and theories. The idea (which I endorse, btw) is that people don't really know much about macroeconomics, and tend to project an unwarranted sense of certitude in their ideas. Of course, Bryan is hardly alone in this belief; Alex Tabarrok famously declared that "a bet is a tax on bullshit".

But this idea, attractive as it is, is not quite true. The reason is something that I've decided to call the Fundamental Error of Risk. It's a mistake that most people make (myself often included!), and that an intro finance class spends months correcting. The mistake is looking at the risk and return of single assets instead of total portfolios. Basically, the risk of an asset - which includes a bet! - is based mainly on how that asset relates to other assets in your portfolio.

This means when people make bets, you don't necessarily know anything about what they really believe. Here is an example. A while ago I made a bet with Brad DeLong that U.S. inflation would go over 5% by 7/28/2015. Brad, who bet against inflation, gave me 50-to-1 odds. Now, if this were my only inflation-related bet, you could infer that I believe that there is a greater than 2% chance of 5% inflation between now and 7/28/2015. But you cannot infer that. In fact, my bet with Brad reveals nothing whatsoever about my inflation beliefs.

Why? Because I also made the exact opposite bet (i.e. that inflation would stay under 5%) with Patrick Chovanec, and gave him only 25-to-1 odds! In other words, I can't possibly lose money, no matter what inflation does (if pizza bets could scale perfectly, I could have executed an arbitrage, but I didn't bother; as things stand, I either break even or win 25 pizza dinner equivalents).

So we see that a bet does not reveal beliefs, because a bet is often used as a hedge. To use another example, I might bet on Sarah Palin winning the presidency, in order to partially hedge my personal sadness in that unfortunate state of the world.

Actually, if you take modern portfolio theory seriously - if you don't believe in any sort of mental accounting at all - then you'd have to look at my entire financial portfolio in order to determine what I really believe about inflation. Even had I not made the countervailing bet with Patrick, I might have been net long in nominal debt (i.e., I might have some cash in a bank account), meaning that my bet with Brad might have just been a hedge against my overall inflation risk.

This is definitely a problem that crops up in finance experiments. Experimentalists try to measure subjects' beliefs about asset price changes by asking them to make side bets about those changes. But we have to be careful to make sure that subjects can't use those side bets as a hedge against their choices in the other parts of their experiment. (You can do this by designing the payoffs such that it's optimal for people to bet on their true beliefs in the side bet, or you can randomly assign people to "prediction" and "investing" groups).

So we see that bets are not necessarily taxes on bullshit. This only becomes more apparent when we bring non-monetary payoffs into the picture. In reality, people make public bets based on all kinds of considerations other than financial gains - ego, fun, the need for posturing, or an excuse to go out for pizza. It's definitely not clear how these other payoffs interact with the monetary payoff of the bet, or with the payoffs of a person's other asset holdings (including other opportunities for ego, posturing, fun, and pizza dinner!). For example, in my case, I made the bet with Brad largely to help publicize the fact of low inflation, and to have an excuse to go to the excellent Zachary's Pizza. And I made the countervailing bet with Patrick not to hedge the risk of a $20 loss, but so I could write blog posts like this. (Also because I am highly mercurial and whimsical.)

Tyler Cowen summed it up best when he tweeted: "I say portfolios reveal beliefs, bets reveal personality traits and public posturing." Exactly.

31 comments:

Your first point does not make any sense at all. You have essentially exited the bet via arbitrage, and instead act as a middle man between two people who are actually making bets on their beliefs: the one who believes the chance of inflation less than 2% and one who believes that the chances are greater than 4%. The market has converged at 3% (with you capturing a 1% discrepancy in both directions in their opinions). It could be concluded that you either believe there is a 3% chance or you no longer hold any opinions about inflation. Either way, it is meaningless, as you are no longer betting on it.

Your second point on betting on Palin as a way to hedge on your utility is meaningless, because this bet could not possibly be big enough to distort (1) the prices of a market or (2) your own opinion expressed in other bets. To stick with your example, there was well over $250 million bet on the Presidential election in 2012 in online betting markets. Even if you bet $10k because your utility was vitally important, that would barely be a ripple in the market. It's just not big enough. And if you did bet $10k, then the rest of your portfolio must be far bigger, and any actual opinion on the election results would probably be expressed in there. If you did not have an opinion expressed elsewhere, then you aren't actually making any 'meaningful' bets, because hedging your utility is a pittance sum.

Finally, your opinion that the portfolio as a whole is a better snapshot of people's true opinions on inflation is a worthwhile one, but it is misleading. The first assumption is that this imaginary inflation-betting market is too small to matter. And the second is that your true opinion is reflected in some secret secondary portfolio. If people are backing up their opinions with bets, it would necessarily be reflected in their ENTIRE betting. You are trying to separate out the portfolio as some secondary thing when it does not make sense to do that.

As to your conclusion: "So we see that bets are not necessarily taxes on bullshit." No, what we have seen is your inability to draw correct conclusions from your own analogies.

"I say portfolios reveal beliefs, bets reveal personality traits and public posturing." Portfolios are bets!!!!! More convoluted, harder to dissect, but bets nonetheless. You need to get past this mental stumbling block.

If you told me that a bunch of wheat farmers had short positions in wheat futures I would think it's more likely that they expect wheat prices to go down than if you told me they hadn't any bets on wheat prices (excluding their production of wheat) or than if you told me they had long positions.

It's clear that the informational content of bets < than the informational content of portfolios. People really may overestimate the informational content of bets, and I think it’s fair to point that. But the informational content of bets > 0.

And also, a call to make bets can be seen as a call to make more portfolio information public. To make more bets in public sphere as opposed to the private sphere, and make your believes more public.

If you told me that a bunch of wheat farmers had short positions in wheat futures I would think it's more likely that they expect wheat prices to go down than if you told me they hadn't any bets on wheat prices (excluding their production of wheat)

Ahh, and there's where you'd make a mistake.

Situation 1: Wheat farmers produce a lot, in part because they expect the price of wheat to go up. However, they hedge against a fall in the wheat price by taking a short position in futures.

Situation 2: Wheat farmers produce very little, because they expect wheat prices to go down. Having produced little wheat, they have little need to take short positions in wheat futures.

In fact, these are the typical situations when it comes to un-diversified commodity producers. Hence, your "Bayesian" rule for inferring beliefs would move your posterior farther from the truth than your prior! You'd be snared by an incorrect highly informative hyper-prior!

It's clear that the informational content of bets < than the informational content of portfolios. People really may overestimate the informational content of bets, and I think it’s fair to point that. But the informational content of bets > 0.

The informational content of bets is highly sensitive to the underlying model...use a misspecified model, and you will glean misinformation from observations of bets, not information!

"The informational content of bets is highly sensitive to the underlying model"

I agree completely.

You're fighting the good fight trying to show the world things are more complicated than they look. But your post could be read as implying there was no information content in bets. As you've just said the true is more complex and interesting.

Oooops! Judging by your reply, my comment must really has some "negative charge". Surely, that was not my intention!

What I was trying to say is the following: even if "all components of a portfolio are publicly observed" I think that we still can't infer the underlying beliefs without making some strong assumptions about the "Homo Economicus" nature of the decision making process. Otherwise, that same portfolio could be the result of so many different beliefs!P.S: Thanks for the advice ;)

I imagine both Brad and Patrick (and their estates, should either pass away) are pretty rock-solvent in terms of pizza delivery.

But perhaps after seeing your above comment one of them will welch for the sole purpose of highlighting the concept of counterparty risk.

There is no such thing as 100% security when a future payoff is dependent upon a counterparty. And in a world with roughly $600 trillion in notional derivatives maybe there will always be a central bank which stands ready to bailout the AIGs of the world. But maybe not.

I'm sorry, but I have to fault your judgement here. If you were angling to eat good Chicago-style pizza, there are a number of places you can get that in Chicago. Eating at Zachary's should be considered a form of punishment. Someone would literally have to pay me to eat the pizza there.

Personally, if your intent was an excuse to eat good food, I think you should have made the bet in bowls of ramen; there's good ramen in the Bay Area--much more than there is good pizza.

Good point, Noah. The prices in the market, or of the bookie, might not reflect what people think the odds are very well because of risk aversion.

Suppose, a person thinks an asset has a 50% probability of a $200,000 payoff and a 50% probability of a negative $100,000 payoff. Even though he thinks the odds are 50-50, he might only be willing to pay $1,000 for the asset, or even a negative amount if a $100,000 loss would be really catastrophic for him. So, you can't infer his believed odds from a straightforward look at his bid price. You'd also have to know his risk aversion.

I thought about this before when it came to betting on very unlikely events. Suppose an event has a probability of 500:1. In the market place it might be hard to find people to take the side of the bet for it not happening, for many dollars.

A lot of people may be fine with putting $100 on it happening. If they lose they just lose $100, not that risky. And if they win, they get $50,000.

But not a lot of people are going to want to bet $50,000 on it not happening, and if they win they make $100, but they risk $50,000 if they lose. Risk aversion makes the two sides of this bet very unequal, even though both parties have the same belief about the odds.

With the lotto you can make it work with massive risk pooling, but I don't think you can find hundreds or thousands of independent super high odds bets you can pool your money over, and with reasonable transactions and evaluation costs. So these very unlikely event markets may far from imply the beliefs of the market participants about the odds.

It seems you missed the other problem with Bryan Caplan's argument: he's basically asking everyone to make unconditional bets based on conditional forecasts. For example, suppose I believe 100% that krugman-style New Keynesian models are exactly correct, and that we are in a liquidity trap. That does not mean that I should bet that inflation would be less than 5% in 2015, because Krugman's actual beliefs are entirely conditional: inflation will be less than 5% given that there is no major stimulus that pushes unemployment below the natural rate, or major asteriod strike that pushes the natural rate above the current unemployment rate. There could be any number of unexpected supply shocks or demand shocks that would contradict the unconditional <5% prediction but that in no way contradict what Krugman "knows" about macro.

You can, of course, also assume priors about the probability of massive stimulus or asteroid strikes, but these are about political science and astronomy, and have nothing to do with your knowledge of economics. Or you could make a conditional bet, such as betting that inflation will be low given no asteroid strikes, but then you have the possibility that neither side wins the bet, if an asteroid actually does hit.

The point is that a persons lack of confidence in a particular bet is not at all indicative of a secret lack of confidence in the model. Even if he is certain of the model, there are still uncertainties in the world.

And this is why if you want to find out about beliefs of person A, you do not ask for a bet, but instead ask for person A to make an offer of a bet for and against an statement, after which you decide which way you take the bet. From the offered odds beliefs and confidence/risk aversion can be deducted.

But the point of the bet is not to "know" the beliefs of someone, but to embarrass his claim publicly by losing that bet. If you bet on both sides, you are signalling that you do not know anything on the matter, and that your beliefs are not trustful neither serious.The issue is not about the money (you can be rich and lose whatever bet you wish to make), but on the public disgrace of having lost. It's the same psychological effect of someone who publicly are noted that he bought something at double the real price.

Public bets can be problematic. For instance, say we bet that Krugman will never apologize to Salim Furth for strongly endorsing Sheldon Whitehouse's factually inaccurate (to the point of hilariously suggesting Irish GDP would decline 95%) attack on his presentation. Obviously if Krugman hears of the bet, he now controls the bet's outcome.

This example is just for fun, but it's an effect too often neglected when talking about things like Fed action, where there are real feedbacks based on how people are betting the future will look according to factors under the control of people who may be influenced by those bets.